Urbanization and development: a Spatial Framework oF rUral-to-Urban migration

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Marco Baudino

GREDEG WP No. 2019-12https://ideas.repec.org/s/gre/wpaper.html

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Urbanization and development: A spatial framework ofrural-to-urban migration†

Marco Baudino‡

GREDEG Working Paper No. 2019–12

Abstract

This paper relies upon some of the assumptions of the classical Alonso-Muth-Mills model in order to construct aspatial framework of rural-to-urban migration; specifically, it develops a spatial framework of migration where ruralworkers are uniformly distributed throughout a rural area, which it develops around a monocentric nuclear urbanarea. The spatial interactions between the rural and the urban areas are modeled via the two spatial variables ofthe rural rents and productivity spillover, whose effects of propagation from the urban to the rural area depend onthe distance of the rural area from the urban area. From the model, it emerges how the rural rents affect the finallevels of congestion in both the two areas, so that the urbanization level of the market solution can be inferior orsuperior with respect to the urbanization level set by the city planner. On the other hand, the inclusion of spatialvariables does not seem to produce scenarios for urban growth which significantly differ from the ones detectedin previous studies. Ultimately, these findings suggest the need for a city planner to design policies affecting thelevel of rural rents in order to modify the desired level of rural-to-urban migration, and hence the desired trade-offbetween urban growth and congestion.

JEL-codes: R12; R23; O43.Keywords: Alonso-Muth-Mills model, Spatial analysis, Migration.

†This work has been supported by the French government, through the UCAJEDI Investments in the Future project managed by theNational Research Agency (ANR) with the reference number ANR-15-IDEX-01.

‡Université Côte d’Azur, CNRS, GREDEG (France). Email:|marco.baudino@univ-cotedazur.fr. Ph. N.ber: +33(0)493954271.

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mailto: marco.baudino@univ-cotedazur.fr

1 Introduction

The effect of urbanization on urban growth in developing countries represents a topic which has been widely analyzedin the literature. It is now a widely accepted fact, among economists, that in less developed economies, urbanizationconstitutes one of the main factors boosting the process of urban development, beside other factors such as capitalaccumulation and quality of the institutions (Turok and McGranahan, 2013[20]; Brülhart and Sbergami, 2009[6];Henderson and Wang, 2009[14]; Henderson, 2003[11]).

As stated by Anderson (2005)[10], in the early stages of growth, economic development is characterized byurbanization, which can be defined as a spatial transformation of the economy, where the population moves throughmigration from an agricultural, rural-based existence, to one where production occurs in cities of endogenous numbersand size. In this process, public and local authorities usually control land markets and regulate migration patterns, andall this clearly affects the city formation progression, as well as the city size. As well, the role of national governmentpolicies regarding trade, labor market and investments in facilities and infrastructures seem to exert a relevant role indetermining the outline of the progressing of the urban setting.

Some authors (Jedwab et al., 2014[15]; Rahman et al., 2006 [1]) further suggested the relevance of the speed ofthe process of urbanization over time. As a matter of fact, urbanization stimulates growth, but the more gradual andcontrolled will be the process of urbanization, the higher will be the likelihood of having higher standards of livingamong the population during the transition phase; in fact, it has been the case that countries which urbanized toofast and without enough control, eventually experienced higher negative externalities (Jedwab et al., 2014[15]).

Since the seminal contribution of Lewis (1954)[16], subsequent general-equilibrium models developed in the liter-ature have tried to frame the long-run relationship between migration, human capital accumulation and urban growthwithin a rural-urban framework (see, e.g., Fan and Stark, 2008[8]; Bertinelli and Black, 2004[3]; Handerson and Wang,2004[13])1; nonetheless, most of these studies did not consider the spatial properties of the urban or rural areas, treat-ing the latter as merely abstract locations. On the other hand, one of the emerging challenges posited in the urbaneconomic literature has been the formulation of a rigorous economic explanation for a variety of observed regularitiesin the spatial structures of real-world cities (Brueckner, 1987[5]). Intuitive examples are represented by the averagedecrease in house size and average increases in the price for land when approaching the city center. These two simpleexamples entail the presence of substitution effects, which, to the best of our knowledge, have virtually never beenconsidered in theoretical models of rural-to-urban migration. Per contrary, in the urban economic literature, sincethe seminal contributions of Alonso, 1964[2], Muth, 1969[18] and Mills, 1967[17], modeling the size and the structureof the urban environment has become a powerful tool to take into account substitution effects, thus providing amore truthful depiction of reality. Specifically, in the simplest framework of the Alonso-Muth-Mills model, the city isspatially represented by a monocentric Central Business District (CBD) around which the rest of the urban area isdeveloped. Urban dwellers travel to the CBD to work, and in order to reduce commuting time, they would choose,ceteris paribus, to live closer to the city center. This inevitably increases the price of the houses, which becomescompensated by a minor housing surface (and this help explaining, for instance, why skyscrapers are generally locatedcloser to city centers). In the end, the overall city size and structure will be simultaneously affected by the populationsize, commuting costs and the cost of the land.

In light of these considerations, we aim to contribute to the literature by developing a spatial framework of rural-to-urban migration drawing upon some key assumptions of the Alonso-Muth-Mills model; this has the main advantageof allowing to go beyond the classic assumption which considers the urban and rural areas as pure abstract locations.

1For a detailed literature review on rural-to-urban migration models, see Henderson, 2004[12].

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Specifically, we model the size and the structure of the rural-urban framework by introducing a monocentric urban areasurrounded by a rural area. We rely upon the (non-spatial) models developed by Handerson and Wang (2004)[13];Bertinelli and Black (2004)[3] and Black and Henderson (1999)[4] to study the relationship between substitutioneffects, migration, human capital accumulation and urban growth. Specifically, we draw upon Handerson and Wang(2004)[13] by introducing the substitution effect of land rents in the rural area. The latter, in the Alonso-Muth-Millsmodel, have been proven to exert a significant impact on urban dwellers’ decision to relocate to the CBD. Adoptingthe same logic, rural rents are likewise expected to affect the decision of rural dwellers to migrate to the urban area.Then, following Black and Henderson (1999)[4], we allow for the presence of a productivity spillover originating in theurban area, and whose effects spatially propagate throughout the rural area. The main reason why we account for aproductivity spillover is to consider the so called radiation effect (QI Jin-li, 2003[19]), which basically consists of urbanimprovements in science, education, health care, etc., that tend to produce positive externalities in the surroundingrural areas, thus spreading knowledge and technological diffusion. Finally, we follow Bertinelli and Black (2004)[3] toderive the optimal share of the urbanization level, as well as to study the evolution of the urban growth process overtime.

The rest of the paper is organized as follows: Section 2 outlines the model, where we discuss the spatial propertiesof the urban and the rural areas. Subsequently, we analyze the optimal levels of urbanization conditioned upon thesubstitution effects exerted by the rural rents and the productivity spillover. Then, we study the long-run process ofurban growth conditioned on human capital accumulation and level of urbanization. Finally, section 3 concludes.

2 Model

2.1 Spatial properties of the urban and rural areas

As specified above, we draw upon the Alonso-Muth-Mills model in order to construct a simple spatial framework ofrural-to-urban migration. To this aim, consider a settlement composed by a monocentric urban area (the CentralBusiness District, or CBD) surrounded by a rural area. The total population of the settlement is expressed by Ψ̄(> 0), with δt (0 ≤ δt ≤ 1) denoting the fraction of the population living in the urban area at time t, and (1− δt) thefraction of the population living in the rural area. The population is composed by individual workers living in the timeperiod t, who decide their amount of investment in human capital at the beginning of the period2. In the rural area,we additionally assume absence of human capital, due to the presence of low skilled tasks requiring an inferior amountof knowledge. Always in the rural area, the gross income of workers depends on κr (> 0), expressing the exogenousproductivity factor of the rural area3 and πt (> 0), the overall initial level of technology for the whole economy attime t. In addition, the gross income of rural workers is influenced by a productivity spillover effect deriving fromthe CBD, which propagates throughout the rural area, and whose effect decreases with the distance d from the urbanarea4. Besides the distance, the final effect of the productivity spillover on the rural income will also depend on theabsolute level of ω (≥ 0), measuring the decay of the spillover itself; that is, when ω increases, the spillover benefit

2Following the literature, for the sake of simplicity, no physical capital is included in the model.3The exogenous productivity factor includes, for instance, the level of the rural infrastructures, the efficiency and quality of the public

utilities, etc.4Namely, the parts of the rural area closer to the CBD will benefit more from the spillover effect compared to the more distant ones. It

is possible to think about the productivity spillover effect as an indirect economic impact exerted by the industrial urban activities towardsthe rural area; such an impact could be expressed, for instance, by the relocation of urban industrial facilities, job creations, improvementsin the rural infrastructures, and so on.

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in the rural area will go down, thus leading to a worsening in the income condition of rural workers; on the contrary,when the decay of ω is flatter, rural workers will benefit a higher beneficial effect from the productivity spillover, thusenjoying higher rural salaries5. Finally, rural workers face a rent for their house given by r(d) (≥ 0), representing theland rent for each unit of occupied living surface they have to pay6.

The equation for the rural income hence writes as follows:

yrt = κrπt − ω(d)− r(d) (1)

Rural workers living in the rural area will then maximize their income with respect to the distance:

maxd(d)

yrt = maxd(d)

[κrπt − ω(d)− r(d)L]

which leads to:

r′(d) = −ω′(d)

The above equation entails that the rent is decreasing with respect to ω(d), in order to compensate a loss in therural income from a lower productivity spillover as we approach the farthest regions of the rural area. Alternatively,more sensible the decay of the spillover effect is with respect to the distance from the CBD, the higher will be therents in getting closer to the CBD itself.

Subsequently, to investigate the geometric properties of the settlement, assume for the sake of simplicity that theproductivity spillover is linear with respect to the distance (i.e., ω(d) = ωd), so that also the gradient of the rentwill be linear. In addition, further assume an homogeneous linear distribution of the rural population around themonocentric urban area, along two segments of equal length. As depicted in Figure 1, the urban area can thereforebe represented by a nuclear CBD, while the two segments of equal length, [d̄, CBD] and [CBD, d̄] represent the lineardistribution of the rural area around the CBD. The two d̄ denote, respectively, the left and the right extremes of therural area, i.e., the two farthest rural regions from the CBD.

Figure 1: Linear distribution of the rural area.

The distribution of rural workers across the rural area will hence be given by the following equation:

d̄ = ((1− δt)Ψ̄)/2

Given then the linearity of the productivity spillover effect and, accordingly, of the gradient of the rents, it ispossible to derive the following expression:

5Here, the terms income, salary and earning are considered interchangeable.6Even if rural workers own the house, because of living inside of it, they always face an opportunity cost given by a potential rent they

loose from not renting the house to other workers.

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r(d) = r(0)− ωd

where r(0) represents the rent in the closest rural region to the CBD, and r̄ the rent at d̄, so that we have:

r(d̄) = r̄ = r(0)− ω(d̄)

from which:

r(0) = r̄ + ω((1− δt)Ψ̄/2)

and therefore:

r(d) = r̄ + ω((1− δt)Ψ̄/2)− ωd

After some algebric manipulations, the equation for the rural income can be rewritten as follows:

yrt = κrπt − r̄ − ω(1− δt)Ψ̄/2 (2)

From (2), some preliminary facts emerge. Firstly, the rural income is increasing in the gross earning but decreasingin r̄; the increase in the opportunity cost for the land brings in fact to an increment of r(d), whatever is the distancefrom CBD. As for ω, the latter, as it was easy to expect, decreases the rural income in increasing the distance fromthe CBD (moreover, as already stressed, an increase in the sensibility of ω(d) to the distance increases the gradientof the rent). Finally, the rural income results to be increasing in the number of workers living in the urban area, anddecreasing in the number of workers living in the rural area; indeed, as more workers remain in the rural area, thehigher the negative effect exerted by the rents in the rural area results to be, due to a higher competition for the landamong rural workers.

When considering the urban area, assume conversely that here, workers have access to more sophisticated pro-duction technologies; hence, they can gain investments in human capital through investments in education (whichhowever require a certain fixed amount of effort). Rural workers are free to relocate from the rural to the urban area(we assume no restrictions on rural-to-urban migration) and invest as well in human capital. At the same time, anexcessive displacement of rural workers in the urban area imposes the problem of congestion externalities due to anexcessive increase in the number of the urban population7. This inevitably generates negative spillovers (for instance,a worsening in the urban level of health conditions, public services and so on) which decrease the general well-beingof urban citizens.

In light of these considerations, the equation for the urban income writes as follows:

yut = κuπt[st]

γ − [ϕst + qδtΨ̄] (3)

Where the first term on the right hand side of (3) represents the gross earning received by a urban worker (beingalready a urban citizen or a rural migrant who decided to relocate to the urban area), with κu expressing the exogenous

7The issue of congestion mainly affects urban areas (as the latter generally face strict space constraints), whereas rural areas are usuallynot (or remarkably less) affected by (severe) issues of congestion provoked by overpopulation. This is why congestion externalities wereonly modeled in the urban area and not in the rural area.

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productivity factor of the urban area. st is the amount of human capital owned by the urban worker at time t, withthe parameter γ (0 < γ < 1) denoting decreasing returns to education. ϕ (> 0) is the effort paid by the worker toobtain the amount of human capital he/she owns. Finally, the last term takes into account the congestion effect ofover urbanization, with q being a positive constant8. In the urban area, workers maximize their income given theiroptimal level of human capital investment; deriving the FOC from (3) with respect to st, yields the optimal level ofhuman capital for a urban worker:

s̃t = (γκuϕ−1πt)

1/(1−γ) (4)

Following Harris and Todaro, 1970[9], we assume that rural-to-urban migration is driven by the gap in the netincome between urban and rural workers. Supposing an initial gap between a superior urban income and a lowerrural income, migration of rural workers towards the urban area will continue up to the point when an equilibrium isachieved. In the end, at this equilibrium level, the rural income will be equal to the urban income. Hence, we willhave the following expression:

κrπt − r̄ −ω

2Ψ̄ +

ω

2δtΨ̄ = κ

uπts̃tγ − ϕs̃t − qδtΨ̄

Once this equilibrium condition is achieved, there will be no migration anymore.

2.2 Optimal levels of urban congestion

From the equation capturing the equilibrium condition among the urban and the rural incomes, it is possible to derivethe fraction of the population δ̂t which eventually locates to the urban area at time t according to the market solution:

δ̂t =2(κuπts̃t

γ − ϕs̃t − κrπt + r̄) + Ψ̄ωΨ̄(2q + ω)

(5)

An investigation on the conditions of existence for δ̂t is necessary. Analyzing the urban and rural incomes, wenotice how their absolute value is influenced by the share of people living in their respective areas. Indeed, it waspreviously demonstrated how the urban income is negatively affected by the congestion costs that increase in thenumber of urban workers9, while the rural income is positively influenced by the decrease in the number of workersliving in the rural area, since this lowers the competition for the land (affecting in turn the rural rents). Therefore,considering expression (3) for the urban income, we see that the latter reaches its maximum when δ̂t = 0, i.e., whenthere is no urban population, whereas the rural income, from expression (2), reaches its highest when 1 − δ̂t = 0 (orδ̂t = 1), i.e., when there is no rural population. The necessary condition for the existence of the equilibrium at δ̂t isthus provided by the following expression:

κrπt − r̄ −ω

2Ψ̄ < κuπts̃t

γ − ϕs̃t < κrπt − r̄ + qΨ̄

The graph in Figure 2 better helps in understanding this condition. For very low levels of urbanization (i.e., forlevels of δt < δ̂t, whose extreme point at δt = 0 represents the lowest value for the rural income, κrπt − r̄ − ω2N ,and the highest value for the urban income, κuπts̃tγ − ϕs̃t), urban workers enjoy remarkably high incomes, due to a

8We further assume a response one for one of the amount of urban population with respect to changes in congestion costs.9We assume that rural workers relocating to the urban area do not consider congestion costs.

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minimal presence of congestion costs; on the other hand, the incomes in the rural areas are very low, given a highnumber of rural workers competing for the land. At the opposite, for very high levels of urbanization (i.e., for levelsof δt > δ̂t, whose extreme point at δt = 1 denotes the highest value for the rural income, κrπt − r̄, and the lowestvalue for the urban income, κuπts̃tγ − ϕs̃t − qΨ̄), congestion costs are remarkably high (and this lowers the urbanincome), while the rural income results to be high (because of reduced land competition). Two different scenarios canhence take place. In the first scenario, when the urban income exceeds the rural income (i.e., when 0 < δt < δ̂t), ruralworkers will migrate to the urban area, in this way increasing the urbanization level and lowering the urban income.Per contrary, in the scenario where the rural income exceeds the urban income (i.e., when δ̂t < δt < 1) we will assistto a migration of urban workers towards the rural area, and this will decrease the urbanization level. In both the twoscenarios, the differential between the two incomes will adjust the urbanization level accordingly, until the equilibriumpoint at δ̂t is achieved. Here, no migration will occur anymore.

Figure 2: Equilibrium conditions.

Following Wheaton (1974)[21], we subsequently perform comparative static analysis on the equilibrium level ofurban density in order to better evaluate the effect of some parameters of interest. Differentiating from (5) yields:

∂δ̂t∂r̄

> 0,∂δ̂t∂κu

> 0,∂δ̂t∂κr

< 0,∂δ̂t∂ω

> 0

These results are straightforward. Higher productivity factors in the urban and rural area will lead, respectively,to higher and lower rates of migration from the rural area to the CBD. In addition, increases in the level of land rentsand in the decay of the productivity spillover will worsen rural workers’ income, pushing them to migrate to the urbanarea.

Contrarily to rural migrants, the city planner in charge of designing the urban policies will consider the increase

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in congestion costs provoked by an excessive urban population. Subsequently, the city planner will accordingly set, attime t, the best feasible number of urban workers maximizing the overall per capita net output of the economy, ȳt:

maxδt

ȳt = (1− δt)[κrπt − r̄ − ω(1− δt)Ψ̄/2] + δt[κuπts̃tγ − ϕs̃t − q(δtΨ̄)]

Deriving the FOC from the above new expression of equilibrium, we obtain the proportion of workers ¯̄δut that theurban planner will hence find as an optimal solution:

¯̄δut =κuπts̃t

γ − ϕs̃t − κrπt + r̄ + Ψ̄ωΨ̄(2q + ω)

(6)

When comparing (5) with (6), the market solution for the optimal share of urban population will exceed inmagnitude the one of the city planner only under certain conditions. Specifically, ¯̄δut < δ̂t verifies iff κuπts̃tγ − ϕs̃t −κrπt + r̄ > 0; conversely, ¯̄δut > δ̂t occurs iff κuπts̃tγ − ϕs̃t − κrπt + r̄ < 010. We will now discuss more accurately thesetwo different cases.

First case ( ¯̄δut < δ̂t)

This first case arises when the gross urban income net of the cost to acquire human capital exceeds the grossrural income net of the negative effect exerted by the rents in the rural area, which entails: κuπts̃tγ − ϕs̃t > κrπt − r̄;re-expressing this condition in terms of ratio between the shares of the net rural and urban income, we obtain:

κrπt−r̄κuπts̃tγ−ϕs̃t < 1. This condition will lead to

¯̄δut < δ̂t, since the final economic well-being, at the eyes of workers, willresult to be higher in the urban area than in the rural area, provoking necessarily a higher urban agglomeration, sincemore rural workers would have eventually relocated to the urban area.

Second case ( ¯̄δut > δ̂t)

The second condition conversely arises whenever κuπts̃tγ − ϕs̃t < κrπt − r̄, or, in terms of ratio between shares,when κ

rπt−r̄κuπts̃tγ−ϕs̃t > 1. This second case is likely to occur when the economic conditions of the rural area are particularly

prosper and therefore rural workers enjoy considerably high rural salaries. Because of this, they will end up neglectingthe negative effect exerted by the rents (since the latter will likely result to be minimal), so that the final result will bea higher agglomeration level in the rural area. The city planner, who considers instead congestion costs and the levelof the rents, will be able to set a solution by which ¯̄δut > δ̂t to counterbalance the myopic attitude of rural workers.Intuitively, the necessary condition for the emergence of the second case could also derive from a situation of lowerrural incomes, as far as, however, the net urban income results to be remarkably low (because of, for instance, asituation of urban economic downturn), always coupled by a negligence by rural workers in considering rural rents.

Therefore, to summarize, even in the presence of congestion costs associated to overpopulation in the urban area,not necessarily the urbanization level of the market solution will exceed the urbanization level set by the city planner;conversely, the opposite may also occur. This will particularly be the case in the presence of remarkably high ruralsalaries11 compared to urban salaries, or when the urban economic conditions are not thriving (e.g., because of an

10In order for ¯̄δut to be positive, the condition for which Ψ̄ω > 2(κuπts̃tγ − ϕs̃t − κrπt + r̄) must always hold.11This condition likely verifies when the exogenous productivity factor in the rural area increases, for instance because of an improvement

in the quality of the infrastructures; the latter has in fact proven to exert a crucial role in increasing the productivity level in rural areas(see, e.g., Duranton and Turner, 2012 [7]).

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economic recession), or given a high cost of human capital accumulation (for instance, because of the presence ofbureaucratic barriers) and/or low returns from education (due to, for instance, a poor education system). In all thesecases, the role exerted by the rural rents results to be crucial, given the impact exerted on the gross rural income, whicheventually will affect the final choice of location of rural workers. Indeed, when the level of the rents is considerablylow and the gross rural income is higher than the urban income, rural workers will not consider the negative effect ofthe rents, and this will consequently lead to more agglomeration in the rural area. On the other hand, the city planner,who as stressed, takes into account both the effect of urban congestion costs and rents, does not see the issue of apotential mass relocation of rural workers from the rural to the urban area, and therefore can increase the desirableshare of workers in the CBD. Nonetheless, the gap between a higher rural income against a lower rural income couldalso be due to other reasons, for instance, in case of urban economic downturns, because of an increased difficulty inhuman capital acquisition, remarkably low returns to scale in education, etc. In all these cases, increasing the numberof urban workers could potentially help to augment the chances of fostering the urban growth.

Conversely, when the net urban salaries greatly exceed the gross rural salaries net of the values of the rents, thedegree of agglomeration set by the urban planner will result to be higher than the one of the market solution; for thisscenario, there are two feasible explanations; first, it can be the case that the urban area has moderate congestioncosts and registers a consistent economic growth, boosted by a good system of urban infrastructures, high returnsfrom education and low costs of acquisition of human capital, thus attracting workers from the rural area. On theother hand, it may also be the case that in reality the urban net salaries are much lower, due to the presence of highcongestion costs; the problem arises because the latter are not perceived by rural workers, thus providing them thefalse impression that in the urban area, workers truly enjoy higher wages. In all these cases, the city planner will hencenarrow the share of desirable urban population, in order to decrease the congestion costs within the urban area.

2.3 Scenarios of urban growth

After having analyzed the drivers influencing the dynamics of rural-to-urban migration, we then study how the latteraffects the process of long-run urban growth, with particular consideration of the effect exerted by the spatial variablesof the rural rents and productivity spillover. Following the literature, we postulate that urban growth is measured byincreases in the level of technology, πt, in the time period t+1. Specifically, the equation of motion linking the overallinitial level of technology πt to the level of technology in the next time period, πt+1, the degree of urbanization, theaverage level of human capital12, and the parameter ρ providing for δt and st decreasing returns (i.e., 0 < ρ < 1),writes as follows:

πt+1 = max f(πt, [δtst]ρ) (7)

This expression entails that technological progress is induced by human capital accumulation and urbanization.What is crucial for the economy at time t are hence the investments in human capital at time t − 1. The degree ofurbanization13, the other variable entering (7), affects the growth process since the higher the number of urban workers,

12The average level of human capital in the whole economy solely depends on the average level of urban human capital, since in therural area we have postulated no human capital. Indeed, denoting as s̄t the average level of human capital in the whole economy, we havethat: s̄t = (1− δt)0 + δtst. Thus, s̄t = δtst. Moreover, for the sake of simplicity, no depreciation is assumed.

13Specifically, the degree of urbanization affects the growth process in two ways, directly and indirectly. Directly as it appears inequation (7), and indirectly because, affecting the level of technology, it automatically influences the optimal level of human capital chosenby urban workers (as the latter depends on πt), which will eventually end up affecting the final equation of motion. It is possible to see

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the higher the investments in human capital boosting urban growth. In addition, the resulting higher productivitythus generated will attract more rural workers to the CBD, who will invest in turn in human capital, creating in thisway a positive circle. Moreover, these investments will rise according to the increase in the technological level. Inthe end, the urban economy will experience different growth paths depending on the values of the parameters in (7),in primis depending on the level of πt. First, if the initial level of technology is too low, the urban area will remainstuck in a development trap without economic growth, and will be likely to remain with its current technological levelforever, without experiencing urbanization. Nonetheless, if the initial level of technology is sufficiently high, the urbangrowth process will begin; this process will lead to a final steady state of partial or full urbanization, through therespective functions governing the technological path of the economy, πpt+1 and π

ft+1.

To sum up, according to the different dynamic paths that the level of technology {πt}∞t=0 will take, we eventuallyobtain these three following scenarios: no-urbanization, partial urbanization, and full urbanization.

πt+1 =

π0 if δt = 0

πpt+1 if 0 < δt < 1 , πpt+1 > 0 and (δtst)

ρ > πt

πft+1 if δt = 1 , πft+1 > 0 and (δtst)

ρ > πt

where the equations of motion πpt+1 and πft+1 were obtained by plugging expressions (4) and (5) into (δtst)ρ. In

the end, the path of the economy will eventually depend on the magnitude and shape of the same functions of motion.We will now proceed in the analysis of each one of the three scenarios.

First scenario: No urbanization

As hinted above, the start of the urbanization process requires an initial threshold level of technology. For lowerinitial levels of At, the urbanization process will never begin, and the economy will remain stuck in a developmenttrap, with δt = 0 and yrt > yut . Plugging (4) into (3), this will occur when the initial level of technology is14:

π0 ≤

[κr

κu1

1−γ ϕγ

γ−1 (γγ

1−γ − γ1

1−γ )

](1−γ)/γ

In this case, the per capita gross earning for both urban and rural workers will end up to be the same: κrπt =κrπ0,∀t. Graphically, this scenario is represented in Figure 3. In the scenario of no urbanization, the shapes of theπpt+1 and π

ft+1 functions are irrelevant, since the final equation of motion will always lie below the 45◦ line15. In this

scenario, the initial level of technology π0 is too low and the incentives to invest in human capital are not sufficientenough; therefore, the dynamic path of the technological level will always remain below the 45◦ line and the urbandevelopment process will never take place.

this by plugging (δtst)ρ into expression (4), to obtain: st =[γκu(δt−1st−1)

ρ

ϕ

]1/(1−γ), which in turn will impact on the final equation of

motion in (7). Of course, an essential condition for generating growth is that the levels of human capital and degree of urbanization attime t− 1 must be higher than the current level of technology at time t (i.e., (δt−1st−1)ρ > πt).

14Since δt = 0, the values for the rents and the spillover effect (which depend on the distance from the CBD) in the rural income willbe zero.

15πpt+1 will result to be concave iff (1− γ)/γ < 1; at the opposite, for certain low levels of γ, it can also be the case that (1− γ)/γ > 1,and therefore the same function of motion will end up being convex. In latter case, however, nothing would change as long as the functionπft+1 is concave and crosses the function π

pt+1 below the 45◦ line.

10

Figure 3: No Urbanization.

Second scenario: Partial urbanization

The necessary condition for the emergence of urban growth is represented by an initial level of technology abovethe threshold level. In addition, the urban income must be higher than the rural income, in order to boost theurbanization process (which requires 0 < δt < 1); in other words, this entails that the urban income net of the costto invest in human capital results to be superior to the rural income net of the effect exerted by rural rents and theproductivity spillover. In this situation, the equation of the dynamic path will be obtained by plugging expressions(4) and (5) into (7), to obtain:

πpt+1 =

[Θ2/(1−γ)(

−2ϕΨ̄(2q + ω)

+2κuπtΘ

(γ−1)/(1−γ)

Ψ̄(2q + ω)) + Θ1/(1−γ)

2r̄ + Ψ̄ω − 2κrπtΨ̄(2q + ω)

]ρwhere Θ = γκuϕ−1πt.As graphically shown in Figure 4, πpt+1 will be concave, with two possible steady state levels for the technological

path (see Appendix 1).

The first steady state level, π̃1, is unstable. For all the values of πt ranging from 0 up this point, the initial level oftechnology is not sufficiently high to start the urban growth process. As well, the workers’ income and their incentiveto invest in human capital result to be too low. Because of this, the economy will remain stuck in a development trap,and the level of growth of the urban area (represented by the dotted arrows) will converge back to zero. However,if the initial level of technology is equal or superior to π̃1, the urban economy will start growing (boosted by theincreasing rates of investments in human capital and urbanization) until it reaches the steady state level π̃2. At thispoint, the growth rate of investments in human capital and urbanization will stop and the growth level of the urbaneconomy will enter a condition of stable equilibrium16.

16The initial level of technology could also start above π̃2; in this case, however, there will be a downturn in investments in humancapital (as well as in the urbanization rate), which will eventually bring back the equilibrium to the steady state level of π̃2.

11

Figure 4: Partial urbanization.

Third scenario: Full urbanization

Finally, in this last scenario, the urban area experiences full urbanization, with two possible long paths of growthwhich are depicted in Figures 5 and 6, respectively. Specifically, the kink of the dynamic function at the pointπ̃2 of Figures 5 (first sub-scenario) and 6 (second sub-scenario) represents the point where, given a highly enoughtechnological level, the entire population will decide to invest in human capital and will locate to the urban area.

Figure 5: Full urbanization with steady state.

The final equation of motion will be equal to the one of the partial urbanization case, but being now δt = 1 wewill have:

12

πft+1 = Θρ/(1−γ)

In this third scenario, πpt+1 will be convex, whereas πft+1 will either be concave (Figure 5) or convex (Figure 6)

(see Appendix 1). In addition, the slope of πpt+1 will always exceed the slope of πft+1, since the condition ∂π

pt+1/∂πt >

∂πft+1/∂πt is always verified (Proof in Appendix 2).In the first sub-scenario (Figure 5), where πft+1 is concave, we assist to a faster urban growth in the first transition

phase of partial equilibrium (given that ∂πpt+1/∂πt > ∂πft+1/∂πt). Once the economy reaches point π̃2, the speed of

growth decelerates, since the second share of rural workers who move to the CBD is smaller in comparison to thefirst share; as a result, there will be a lower number of new investments in human capital boosting urban growth.Eventually, the economy reaches full urbanization at the new (higher) steady state level π̃3, with per capita earningsequal to [π̃3κu]

11−γ ϕ

γγ−1 (γ

γ1−γ − γ

11−γ )− qΨ̄ (being δt = 1).

Finally, the second sub-scenario (Figure 6) entailing a convex πft+1 function, requires the condition γ + ρ ≥ 1(see Appendix 1). This means that the returns to investments in human capital are remarkably high and/or theeffect of human capital on technology is considerable. In this second sub-scenario, the urban economy and the level oftechnology will grow boundless; in particular, the urban economy will firstly reach the stage of full urbanization (atpoint π̃2), to then continue its path, with per capita earnings (and growth rates) increasing forever.

Figure 6: Full urbanization with unbounded growth.

3 Conclusion

This paper aimed at contributing to the literature by extending the classical models of rural-to-urban migration sofar developed in previous studies by introducing a spatial framework for the urban and rural areas. Specifically, wedeveloped a spatial framework of rural-to-urban migration, relying upon some of the key assumptions of the Alonso-Muth-Mills model. Hence, we defined a settlement composed by a monocentric nuclear urban area surrounded by

13

a linearly distributed rural area. Following the literature, we conditioned urban growth as a function of humancapital (the latter being accumulated only in the urban area), with rural-to-urban migration triggered by the incomedifferential between urban and rural workers. The novelty introduced in this paper was to model the spatial interactionbetween the urban and the rural areas by introducing the spacial variables of the rural rents and the productivityspillover, whose effect on rural workers’ income depend on the distance from the urban area.

With respect to the findings of previous studies, in this model, not necessarily the urbanization level associated tothe market solution exceeds the urbanization level set by the city planner; in this regard, the spatial variable of ruralrents exerts a main impact in affecting the two optimal levels of urbanization. Specifically, when rural workers enjoyconsiderably high salaries, they inevitably end up neglecting the effect exerted by the rural rents and accordingly, themarket solution for the urbanization level will be lower, in absolute value, than the city planner solution, thus leadingto a higher level of agglomeration in the rural area. On the other hand, in presence of higher urban salaries and/orreduced levels of rural rents, rural workers will be attracted to move to the urban area, so that, contrarily to theprevious case, the market solution will now be higher, in absolute value, than the city planner solution. Subsequently,we studied the dynamic evolution of urban growth over time, following the assumption posited in the literature forwhich urban growth is measured by increases in the level of technology, with the variables of the rural rents andthe productivity spillover entering the equations of motion. According to the initial level of technology, differentgrowth scenarios emerged. For too low levels of technology, a development trap could arise, since investments inhuman capital are not sufficiently high. This scenario always entail lower rural incomes compared to urban incomes.Nonetheless, when the initial level of technology is sufficiently high, the urban process begins, leading to an increase inthe technology level and output over time. A necessary condition for the start of the urbanization process is that theurban income must be higher than the rural income; this requires necessarily that the urban income net of the costto invest in human capital results to be superior to the rural income net of the effect exerted by rural rents and theproductivity spillover. When the growth process begins, the final level of steady state will eventually depend on thelevel of returns to human capital and impact of human capital on technology; for remarkably high values of returnsto education, the urban area might also experience unbounded growth. However, these growth benefits deriving fromover- (full) urbanization also entail the emergence of congestion costs, which will ultimately lower the quality of life ofurban workers. Therefore, in the end, when the urbanization level of the market solution exceeds the urbanization levelset by the city planner, the latter will face a trade-off between growth and negative urban congestion externalities;within this context, if the aim of the city planner is to reduce the congestion level of the urban area, it is likelythat policies designed to decrease, for instance, the value of rural rents, could contribute to improve the standard ofliving of rural workers, thus resulting in lower rates or rural-to-urban migration. On the other hand, urban policiesaimed at positively influencing the urban productivity factor (e.g., policies designed to improve the quality of urbaninfrastructures) may help increasing the optimal level of urbanization, thus allowing the relocation in the urban areaof a larger share of rural workers.

14

Appendix 1

Dynamics of πpt+1 and πft+1 functions.

As specified in the main text, the dynamic evolution of the technological level of the economy will depend on theshape of the function πt+1. Specifically:

πt+1 =

π0 if δt = 0

πpt+1 if 0 < δt < 1 , πpt+1 > 0 and (δtst)

ρ > πt

πft+1 if δt = 1 , πft+1 > 0 and (δtst)

ρ > πt

where:

πpt+1 =

[Θ2/(1−γ)

(−2ϕ

Ψ̄(2q + ω)+

2κuπtΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)

)+Θ1/(1−γ)

2r̄ + Ψ̄ω − 2κrπtΨ̄(2q + ω)

]ρand:

πft+1 = Θρ/(1−γ)

Now; πpt+1 < 0 when the following conditions are met: πt <ϕ

κuΘ(γ−1)/(1−γ)∧ πt < 2r̄+Ψ̄ω2κr , or πt <

ϕκuΘ(γ−1)/(1−γ)

∧ πt > 2r̄+Ψ̄ω2κr , with Θ2/(1−γ)(

−2ϕΨ̄(2q+ω)

+ 2κuπtΘ

(γ−1)/(1−γ)

Ψ̄(2q+ω)

)> Θ1/(1−γ) 2r̄+Ψ̄ω−2κ

rπtΨ̄(2q+ω)

. The same

conditions are always satisfied when πt > ϕκuΘ(γ−1)/(1−γ) ∧ πt <2r̄+Ψ̄ω2κr , with Θ2/(1−γ)

(−2ϕ

Ψ̄(2q+ω)+ 2κ

uπtΘ(γ−1)/(1−γ)

Ψ̄(2q+ω)

)< Θ1/(1−γ) 2r̄+Ψ̄ω−2κ

rπtΨ̄(2q+ω)

. Otherwise, the function πpt+1 is strictly positive.

In differentiating πpt+1 with respect to πt, we obtain:

∂πpt+1∂πt

= ρ

[Θ2/(1−γ)

(−2ϕ

Ψ̄(2q + ω)+

2κuπtΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)

)+Θ1/(1−γ)

2r̄ + Ψ̄ω − 2κrπtΨ̄(2q + ω)

]ρ−1×[2Θ2/(1−γ)κuΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)− 2Θ

1/(1−γ)κr

Ψ̄(2q + ω)

]

In the relevant range where πpt+1 and δt are both positive, the first expression in brackets of∂πpt+1∂πt

is positive. Thesecond expression in brackets of ∂π

pt+1

∂πtis positive as long as B > ( κrκu )

1−γγ , which can be rewritten as πt > ϕγκr

1−γγ ( 1κu )

1γ ,

which is always the case in the relevant range in question. The function reaches its minimum at πt = ϕγκr1−γγ ( 1κu )

1γ ,

and is monotonically increasing thereafter.The evolution of the economy will eventually depend on the curvature of the πt+1 function. In particular, the

number of times that this function intersects the 45◦ line (where πpt+1 = πt) will be a crucial factor for the dynamicpath of the economy. Subsequently, in taking the second derivative of πpt+1, we get:

15

∂2πpt+1∂πt2

= ρ(ρ− 1)[Θ2/(1−γ)

(−2ϕ

Ψ̄(2q + ω)+

2κuπtΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)

)+Θ1/(1−γ)

×2r̄ + Ψ̄ω − 2κrπt

Ψ̄(2q + ω)

]ρ−2[2Θ2/(1−γ)κuΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)− 2Θ

1/(1−γ)κr

Ψ̄(2q + ω)

]2

To simplify the calculations, label Θ2/(1−γ)(

−2ϕΨ̄(2q+ω)

+ 2κuπtΘ

(γ−1)/(1−γ)

Ψ̄(2q+ω)

)= Φ and Θ1/(1−γ) × 2r̄+Ψ̄ω−2κ

rπtΨ̄(2q+ω)

= Ω.

The function πpt+1 will be convex when∂2πpt+1∂πt2

is positive and this happens when both Φ < 0, Ω < 0. Or whenΦ > 0 and Ω < 0, with |Φ| < |Ω|. Or Φ < 0 and Ω > 0, with |Φ| > |Ω|. In these three case, the slope of themonotonically increasing function of motion will be increasing for ∀πt and therefore there will always be a singleintersection point where πpt+1 = πt.

Conversely, πpt+1 will be concave when∂2πpt+1∂πt2

is negative and this happens when both Φ > 0, Ω > 0. Or whenΦ > 0 and Ω < 0, with |Φ| > |Ω|. Or Φ < 0 and Ω > 0, with |Φ| < |Ω|. In these cases, the slope of the monotonicallyincreasing function of motion will be decreasing and therefore there could be zero, one or two values of πt to allowtangency points (i.e., when πpt+1 = πt); depending on the values of the parameters it can hence be the case that for∀πt, πpt+1 < πt and in this case there will not be any tangency point. It could then be the case that the conditionπpt+1 = πt is verified and therefore we have one single crossing point. Or, if π

pt+1 > πt there will be two tangency

points.Finally, for the study of the πft+1 function, the calculations are relatively easier. In differentiating π

ft+1, we get:

∂πft+1∂πt

=ρ

1− γΘ

ρ1−γ −1

The function is hence monotonically increasing. Taking the second derivative, we obtain:

∂2πft+1∂πt2

=ρ

1− γ

(ρ− 1 + γ1− γ

)Θ

ρ1−γ −2

Therefore πft+1 will be monotonically increasing and globally concave if γ + ρ < 1, and monotonically increasingbut globally convex if γ + ρ > 1.

Appendix 2

∂πpt+1/∂πt > ∂πft+1/∂πt.

Proof. Consider the point where δt = 1, and πpt+1 = πft+1. Recalling that:

16

∂πpt+1∂πt

= ρ

[Θ2/(1−γ)

(−2ϕ

Ψ̄(2q + ω)+

2κuπtΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)

)+Θ1/(1−γ)

2r̄ + Ψ̄ω − 2κrπtΨ̄(2q + ω)

]ρ−1×[2Θ2/(1−γ)κuΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)− 2Θ

1/(1−γ)κr

Ψ̄(2q + ω)

]and

∂πft+1∂πt

=ρ

1− γΘ

ρ1−γ −1

at the intersection of the two functions we have that:

∂πft+1∂πt

=ρ

1− γπft+1πt

−1 =ρ

1− γπpt+1πt

−1

Assume now that ∂πpt+1

∂πt>

∂πft+1∂πt

. This entails that:

ρ

[Θ2/(1−γ)

(−2ϕ

Ψ̄(2q + ω)+

2κuπtΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)

)+Θ1/(1−γ)

2r̄ + Ψ̄ω − 2κrπtΨ̄(2q + ω)

]ρ−1×[2Θ2/(1−γ)κuΘ(γ−1)/(1−γ)

Ψ̄(2q + ω)− 2Θ

1/(1−γ)κr

Ψ̄(2q + ω)

]>

ρ

1− γπt

−1[Θ2/(1−γ)(

−2ϕΨ̄(2q + ω)

+2κuπtΘ

(γ−1)/(1−γ)

Ψ̄(2q + t)

)+Θ1/(1−γ)

2r̄ + Ψ̄ω − 2κrπtΨ̄(2q + ω)

]ρSimplifying, we obtain:

(1

1− γ− 1

)2Θ(γ+1)/(1−γ)κu

Ψ̄(2q + ω)+

(1− 1

1− γ

)2Θ1/(1−γ)κr

Ψ̄(2q + ω)+

1

1− γ

[−2Θ2/(1−γ)ϕπtΨ̄(2q + ω)

+Θ1/(1−γ)2r̄ + Ψ̄ω

πtΨ̄(2q + ω)

]< 0

Which leads to:

(πtκu)1/(1−γ)(γϕ−1)γ/(1−γ) − γγ/(1−γ)(πtκu)1/(1−γ)ϕγ/(γ−1)

+γ−1[r̄ +

Ψ̄ω

2

]< κrπt

or:

r̄ + Ψ̄ω2γ

< κrπt

Since yrt ≥ 0 and 0 < δt ≤ 1, given (2), the above inequality is always true. �

17

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IntroductionModelSpatial properties of the urban and rural areasOptimal levels of urban congestionScenarios of urban growth

Conclusion